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Grothendieck-Lidski Theorem for Subspaces of Quotients of Lp-Spaces

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Grothendieck-Lidski Theorem for Subspaces of Quotients of Lp-Spaces Grothendieck-Lidskitheorem for subspaces of quotients of Lp-spaces Oleg Reinov and Qaisar Latif Abstract. Generalizing A. Grothendieck's (1955) and V.B. Lidski's(1959) trace formulas, we have shown in a recent paper that for p 2 [1; 1] and s 2 (0; 1] with 1=s = 1 + j1=2 − 1=pj and for every s-nuclear operator T in every subspace of any Lp(ν)-space the trace of T is well dened and equals the sum of all eigenvalues of T: Now, we obtain the analogues results for subspaces of quotients (equivalently: for quotients of subspaces) of Lp-spaces. In the note [13], we have proved that if p 2 [1; 1] and 1=s = 1+j1=2−1=pj; then for any subspace (or quotient) of an Lp-space and for every s-nuclear operator T in the space the nuclear trace of T is well-dened and equals the sum of all eigenvalues of T: The main fact, we are going to obtain here, is Theorem. Let Y be a subspace of a quotient (or a quotient of a subspace) of an Lp-space, 1 ≤ p ≤ 1: If T 2 Ns(Y; Y )(s-nuclear), where 1=s = 1 + j1=2 − 1=pj; then 1. theP (nuclear) trace of T is well dened, 2. 1 j j 1 where f g is the system of all eigenvalues of the n=1 λn(T ) < ; λn(T ) operator T (written in according to their algebraic multiplicities) and X1 trace T = λn(T ): n=1 Let us mention that in the proof we have to repeat some ideas of proofs from [13] (in particular, of the proof of main lemma there) as well as, simultaneously, to use the main lemma [13] itself (so, we will get a generalization of the lemma by using a part of its proof and also its statement). 1. Preliminaries. All the terminology and facts we use here can be found in [58]. ∗ b Let X; Y be Banach spaces. For s 2 (0; 1]; denote by X ⊗sY the completion of the tensor product X∗ ⊗ Y (considered as a linear space of all nite rank operators) The research was supported by the Higher Education Commission of Pakistan. AMS Subject Classication 2010: 47B06. Key words: s-nuclear operators, eigenvalue distributions. 1 GROTHENDIECK-LIDSKII THEOREM FOR SUBSPACES OF QUOTIENTS OF Lp-SPACES 2 with respect to the quasi-norm ! XN 1=s XN jj jj f jj 0 jjs jj jjs 0 ⊗ g z s := inf xk yk : z = xk yk : k=1 k=1 Let Ψp; for p 2 [1; 1]; be the ideal of all operators which can be factored through ∗ b a subspace of a quotient of an Lp-space. Put Ns(X; Y ) := image of X ⊗sY in the space L(X; Y ) of all bounded linear transformations under the canonical factor map ∗ b X ⊗sY ! Ns(X; Y ) ⊂ L(X; Y ): We consider the (Grothendieck) space Ns(X; Y ) of all s-nuclear operators from X to Y with the natural quasi-norm, induced from ∗ b X ⊗sY: Finally, let Ψp;s be the quasi-normed product Ns ◦Ψp of the corresponding ideals equipped with the natural quasi-norm νp;s: if A 2 Ns ◦ Ψp(X; Y ) then A = ' ◦ T with T = βα 2 Ψp;' = δ∆γ 2 Ns and α β γ ∆ δ A : X ! Xp ! Z ! c0 ! l1 ! Y; where all maps are continuous and linear, Xp is a subspace of a quotient of an Lp-space, constructed on a measure space, and ∆ is a diagonal operator with the diagonal from ls: Thus, A = δ∆γβα and A 2 Ns: Therefore, if X = Y; the spectrum of A; sp (A); is at most countable with only possible limit point zero. Moreover, A is a Riesz operator with eigenvalues of nite algebraic multiplicities and sp (A) ≡ sp (B); where B := αδ∆γβ : Xp ! Xp is an s-nuclear operator, acting in a subspace of a quotient of an Lp-space. Denition. Let Y be a Banach space and s 2 (0; 1]: We say that Y possesses the property APs ( the approximation property of order s; written down as "Y 2 APs") ∗ b if for any tensor element z 2 Y ⊗sY the operator z~ : Y ! Y; associated with z; is zero i the tensor element z is zero as an element of the space Y ∗⊗bY: ∗ b This is equivalent to the fact that if z 2 Y ⊗sY then it follows from trace z ◦ R = 0; 8 R 2 Y ∗ ⊗ Y that trace U ◦ z = 0 for every U 2 L(Y; Y ∗∗): There is a simple characterization of the condition Y 2 APs in terms of the approximation of the identity idY on some sequences of the space Y; but we omit it. We need here only some examples which are crucial for our note. For giving them, we formulate and prove the following statement, which, we hope, is interesting by itself. Proposition 1. Let α 2 [0; 1=2] and 1=s = 1 + α: For a Banach space Y; suppose that (α) there exist constants C > 0 such that for every " > 0; for every natural n and for every n-dimensional subspace E of Y there exists a nite rank operator R α in Y so that jjRjj ≤ Cn and jjRjE − idE jjL(E;Y ) ≤ ": Then Y 2 APs: GROTHENDIECK-LIDSKII THEOREM FOR SUBSPACES OF QUOTIENTS OF Lp-SPACES 3 ∗ b Proof. Suppose that there is an element z 2 Y ⊗sY such that trace z = b > 0; but z~ = 0: Consider a representation of z of the kind X1 0 ⊗ z = µkyk yk; k=1 P where jj 0 jj jj jj and ≥ 1 s 1 Without loss of generality, we can yk ; yk = 1 µk 0; k=1 µk < : P (and do) assume that the sequence is decreasing and that 1 ≤ In this (µk) k=1 µk 1: situation, s so, there are with ! and ≤ 1=s µk = o(1=k); ck > 0 ck 0 µk ck=k : Fix any natural N; large enough, such that for all m ≥ N Xm h 0 i ≥ µk yk; yk b=2: k=1 For such an put f gm and apply the condition to nd a corre- m; E := span yk k=1; (α) sponding operator R 2 Y ∗ ⊗ Y for n = m and " = b=4: By our assumption, trace R ◦ z = 0: From this, we get (for all m ≥ N): Xm X1 h 0 i h 0 i 0 = µk yk; Ryk + µk yk; Ryk : k=1 k=m+1 For the rst sum: Xm Xm Xm h 0 i ≥ h 0 i − h 0 − i ≥ − µk yk; Ryk µk yk; yk µk yk; yk Ryk b=2 b=4 = b=4: k=1 k=1 k=1 For the second sum: Z X1 1 h 0 i ≤ α −1=s ≤ α 1−1=s µk yk; Ryk Cm c~m x dx dm m m = dm; k=m+1 m where 0 ≤ c~m ! 0; and thus 0 ≤ dm ! 0: Now, from the last three relations, we obtain: 0 ≥ b=4 − dm: Let us consider some consequences of the proposition. Corollary 1. Let α 2 [0; 1=2] and 1=s = 1 + α: For a Banach space Y; suppose that there exist constants C > 0 such that for every natural n and for every n- dimensional subspace E of Y there exists a nite rank operator R in Y so that α jjRjj ≤ Cn and RjE = idE : Then Y 2 APs: Corollary 2. Let α 2 [0; 1=2] and 1=s = 1 + α: For a Banach space Y; suppose that there exist constants C > 0 such that for every natural n and for every n- dimensional subspace E of Y there exists a nite dimensional subspace F of Y; α containing E and Cn -complemented in Y: Then Y 2 APs: Corollary 3. Let α 2 [0; 1=2] and 1=s = 1 + α: For a Banach space Y; sup- pose that there exist constants C > 0 such that for every natural n and every α n-dimensional subspace E of Y is Cn -complemented in Y: Then Y 2 APs: More- over, every subspace of the space Y has the APs: GROTHENDIECK-LIDSKII THEOREM FOR SUBSPACES OF QUOTIENTS OF Lp-SPACES 4 It can be shown (but we do not need this in the note) that Y 2 APs i for every ∗ b Banach space X the natural mapping X ⊗sY ! L(X; Y ) is one-to-one (for other related results see, e.g., [11], [12]). Thus, taking this in account, we get: Corollary 4. In all above four assertions, in the case of Y with mentioned ∗ b properties, we have the quasi-Banach equality X ⊗sY = Ns(X; Y ); whichever the ∗ b space X was. In particular, Y ⊗sY = Ns(Y; Y ): Before giving more concrete applications of Proposition 1, let us mention the simplest example. Example 1. Let s 2 (0; 1]; p 2 [1; 1] and 1=s = 1 + j1=p − 1=2j: Any subspace as well as any factor space of any Lp-space have the property APs: We used this example in [13]. The statement of Example 1 follows from Corollary 4 and the results of D.R. Lewis (see [3], Corollary 4). As s matter of fact, one can get from the work [3] more general facts on com- plementability concerning Lp-situation. However, we prefer to consider abstract situations and to deal with spaces of nontrivial types and cotypes (partially, for using the results to be obtained in other considerations).
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